Radial averaging operator acting on Bergman and Lebesgue spaces

It is shown that the radial averaging operator T_{\omega}(f)(z)=\frac{\int_{|z|}^{1}f\bigl{(}s\frac{z}{|z|}\bigr{)}\omega(s)% \,ds}{\widehat{\omega}(z)},\quad\widehat{\omega}(z)=\int_{|z|}^{1}\omega(s)\,ds, induced by a radial weight ω on the unit disc {\mathbb{D}} , is bounded from the weighted Bergman space {A^{p}_{\nu}} , where {0<p<\infty} and the radial weight ν satisfies {\widehat{\nu}(r)\leq C\widehat{\nu}(\frac{1+r}{2})} for all {0\leq r<1} , to {L^{p}_{\nu}} if and only if the self-improving condition \sup_{0\leq r<1}\frac{\widehat{\omega}(r)^{p}}{\int_{r}^{1}s\nu(s)\,ds}\int_{0% }^{r}\frac{t\nu(t)}{\widehat{\omega}(t)^{p}}\,dt<\infty is satisfied. Further, two characterizations of the weak-type inequality \eta(\{z\in\mathbb{D}:|T_{\omega}(f)(z)|\geq\lambda\})\lesssim\lambda^{-p}\|f% \|_{L^{p}_{\nu}}^{p},\quad\lambda>0, are established for arbitrary radial weights ω, ν and η. Moreover, differences and interrelationships between the cases {A^{p}_{\nu}\to L^{p}_{\nu}} , {L^{p}_{\nu}\to L^{p}_{\nu}} and {L^{p}_{\nu}\to L^{p,\infty}_{\nu}} are analyzed.